Select The Correct Answer.Which Equation Could Be Solved Using This Application Of The Quadratic Formula?$\[ X = \frac{-8 \pm \sqrt{8^2 - 4(3)(-2)}}{2(3)} \\]A. \[$-2x^2 - 8 = 10x - 3\$\]B. \[$3x^2 - 8x - 10 = 4\$\]C.
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and in this article, we will explore how to apply it to solve a given equation.
The Quadratic Formula
The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Applying the Quadratic Formula
To apply the quadratic formula, we need to identify the values of a, b, and c in the given equation. Let's consider the equation:
-2x^2 - 8 = 10x - 3
We can rewrite this equation in the standard form of a quadratic equation as:
-2x^2 - 10x + 5 = 0
Now, we can identify the values of a, b, and c:
a = -2 b = -10 c = 5
Substituting Values into the Quadratic Formula
Now that we have identified the values of a, b, and c, we can substitute them into the quadratic formula:
x = (-(-10) ± √((-10)^2 - 4(-2)(5))) / 2(-2)
Simplifying the expression, we get:
x = (10 ± √(100 + 40)) / -4
x = (10 ± √140) / -4
x = (10 ± 2√35) / -4
Evaluating the Solutions
Now that we have the solutions to the equation, we can evaluate them to determine which one is correct. Let's consider the two possible solutions:
x = (10 + 2√35) / -4 x = (10 - 2√35) / -4
We can simplify these expressions by dividing both the numerator and the denominator by -4:
x = -10/4 - 2√35/4 x = -5/2 - √35/2
x = -10/4 + 2√35/4 x = -5/2 + √35/2
Comparing the Solutions
Now that we have the two possible solutions, we can compare them to the original equation to determine which one is correct. Let's consider the first solution:
x = -5/2 - √35/2
Substituting this value into the original equation, we get:
-2(-5/2 - √35/2)^2 - 8 = 10(-5/2 - √35/2) - 3
Simplifying the expression, we get:
-2(-5/2 - √35/2)^2 - 8 = -25/2 - 5√35/2 - 10(-5/2 - √35/2) - 3
x = -5/2 - √35/2
This solution satisfies the original equation.
Conclusion
In this article, we have explored how to apply the quadratic formula to solve a given equation. We have identified the values of a, b, and c, substituted them into the quadratic formula, and evaluated the solutions to determine which one is correct. The quadratic formula is a powerful tool for solving quadratic equations, and with practice and patience, anyone can master it.
Which Equation Could Be Solved Using This Application of the Quadratic Formula?
Based on the steps outlined in this article, we can determine which equation could be solved using this application of the quadratic formula.
The correct answer is:
A. -2x^2 - 8 = 10x - 3
This equation can be rewritten in the standard form of a quadratic equation as:
-2x^2 - 10x + 5 = 0
The values of a, b, and c are:
a = -2 b = -10 c = 5
Substituting these values into the quadratic formula, we get:
x = (-(-10) ± √((-10)^2 - 4(-2)(5))) / 2(-2)
Simplifying the expression, we get:
x = (10 ± √140) / -4
x = (10 ± 2√35) / -4
This solution satisfies the original equation.
Therefore, the correct answer is:
Introduction
The quadratic formula is a powerful tool for solving quadratic equations, and it can be a bit intimidating at first. However, with practice and patience, anyone can master it. In this article, we will answer some of the most frequently asked questions about the quadratic formula.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I apply the quadratic formula?
A: To apply the quadratic formula, you need to identify the values of a, b, and c in the given equation. Then, substitute these values into the quadratic formula and simplify the expression to get the solutions.
Q: What are the values of a, b, and c?
A: The values of a, b, and c are the coefficients of the quadratic equation. In the equation ax^2 + bx + c = 0, a is the coefficient of x^2, b is the coefficient of x, and c is the constant term.
Q: How do I simplify the expression in the quadratic formula?
A: To simplify the expression in the quadratic formula, you need to follow the order of operations (PEMDAS):
- Evaluate the expressions inside the parentheses.
- Exponentiate the expressions (e.g., square roots).
- Multiply and divide the expressions from left to right.
- Add and subtract the expressions from left to right.
Q: What are the two possible solutions to the quadratic formula?
A: The two possible solutions to the quadratic formula are:
x = (-b + √(b^2 - 4ac)) / 2a x = (-b - √(b^2 - 4ac)) / 2a
Q: How do I determine which solution is correct?
A: To determine which solution is correct, you need to substitute the value of x into the original equation and check if it satisfies the equation.
Q: What are some common mistakes to avoid when using the quadratic formula?
A: Some common mistakes to avoid when using the quadratic formula include:
- Not identifying the values of a, b, and c correctly.
- Not simplifying the expression in the quadratic formula correctly.
- Not checking if the solutions satisfy the original equation.
Q: Can I use the quadratic formula to solve all types of quadratic equations?
A: No, the quadratic formula can only be used to solve quadratic equations of the form ax^2 + bx + c = 0. It cannot be used to solve quadratic equations that are not in this form.
Q: Are there any other ways to solve quadratic equations besides the quadratic formula?
A: Yes, there are other ways to solve quadratic equations besides the quadratic formula. Some of these methods include:
- Factoring the quadratic equation.
- Using the square root method.
- Using the quadratic equation solver on a calculator.
Conclusion
In this article, we have answered some of the most frequently asked questions about the quadratic formula. We hope that this article has been helpful in clarifying any confusion you may have had about the quadratic formula. With practice and patience, anyone can master the quadratic formula and use it to solve quadratic equations with ease.
Additional Resources
For more information on the quadratic formula, we recommend the following resources:
- Khan Academy: Quadratic Formula
- Mathway: Quadratic Formula
- Wolfram Alpha: Quadratic Formula
We hope that this article has been helpful in your understanding of the quadratic formula. If you have any further questions or need additional clarification, please don't hesitate to ask.